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Abstract
The eigenvalue complementarity problem (EiCP) differs from the traditional eigenvalue problem in that the primal and dual variables belong to a closed and convex cone K and its dual, respectively, and satisfy a complementarity condition. In this paper we investigate the solution of the second-order cone EiCP (SOCEiCP) where K is the Lorentz cone. We first show that the SOCEiCP reduces to a special Variational Inequality Problem on a compact set defined by K and a normalization constraint. This guarantees that SOCEiCP has at least one solution, and a new enumerative algorithm is introduced for finding a solution to this problem. The method is based on finding a global minimum of an appropriate nonlinear programming (NLP) formulation of the SOCEiCP using a special branching scheme along with a local nonlinear optimizer that computes stationary points on subsets of the feasible region of NLP associated with the nodes generated by the algorithm. A semi-smooth Newton's method is combined with this enumerative algorithm to enhance its numerical performance. Our computational experience illustrates the efficacy of the proposed techniques in practice.
Acknowledgements
The authors also thank two anonymous referees for their constructive and insightful comments.
Disclosure
No potential conflict of interest was reported by the authors.
Funding
The research of Masao Fukushima was partially supported by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science. The research of Luís M. Fernandes and Joaquim J. Júdice was partially supported in the scope of R&D Unit UID/EEA/50008/2013, financed by the applicable financial framework (FCT/MEC through national funds and when applicable co-funded by FEDER – PT2020 partnership agreement.)